论文标题
PDE-LEARN：使用深度学习从嘈杂，有限的数据中发现部分微分方程
PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data
论文作者
论文摘要
在本文中，我们引入了PDE-Learn，这是一种新型的深度学习算法，可以直接从嘈杂的，有限的物理系统系统中识别偏差方程（PDE）。 PDE-LEARN使用一个理性的神经网络$ U $来近似系统响应功能和稀疏，可训练的向量，$ξ$，以表征系统响应功能满足的隐藏PDE。我们的方法使用（1）使$ u $近似系统响应功能的损失功能对$ u $和$ξ$的培训融合，（2）封装了以下事实：$ u $满足$ξ$表征的隐藏的pde，（3）使用$ξ$促进$ξ$使用迭代重新启动的最小值squares的想法来促进$ξ$。此外，PDE-Learn可以同时从几个数据集中学习，从而使其能够合并多个实验的结果。这种方法产生了一种强大的算法，可以直接从现实的科学数据中发现PDE。我们通过识别噪声和有限测量值的几个PDE来证明PDE-Learn的功效。
In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational Neural Network, $U$, to approximate the system response function and a sparse, trainable vector, $ξ$, to characterize the hidden PDE that the system response function satisfies. Our approach couples the training of $U$ and $ξ$ using a loss function that (1) makes $U$ approximate the system response function, (2) encapsulates the fact that $U$ satisfies a hidden PDE that $ξ$ characterizes, and (3) promotes sparsity in $ξ$ using ideas from iteratively reweighted least-squares. Further, PDE-LEARN can simultaneously learn from several data sets, allowing it to incorporate results from multiple experiments. This approach yields a robust algorithm to discover PDEs directly from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by identifying several PDEs from noisy and limited measurements.
